Question

Determine whether each table or equation represents an inverse or a direct variation. Explain.

$x$	1	2	5	6
$y$	30	15	6	5

Short answer

The table represents an inverse variation.

Step-by-step solution

Step 1. State the concept of direct variation and inverse variation.

The relationship of the form$y=kx$ is called direct variation.

Relationship of the form $xy=k$ or $y=\frac{k}{x}$ where $x,y\ne 0$ for some nonzero constant $k$ is known as inverse variation i.e., $y$varies inversely as $x$.

Step 2. Check for inverse variation. 

Find $xy$each ordered pair from the table and get,

$\begin{array}{c}1\cdot 30=30\\ 2\cdot 15=30\\ 5\cdot 6=30\\ 6\cdot 5=30\end{array}$

The product is constant so the table represents an inverse variation, $xy=30$.

Step 3. Check for direct variation.

Find $\frac{y}{x}$each ordered pair from the table and get,

role="math" localid="1648108929404" $\begin{array}{c}\frac{30}{1}=30\\ \frac{15}{2}=7.5\\ \frac{6}{5}=1.2\end{array}$

$\begin{array}{c}\frac{5}{6}=0.833\end{array}$

Since $\frac{y}{x}$is not constant so the table doesn’t represent the direct variation.